The Microeconomics of Complex Economies || Tools I

C H A P T E R

2

Tools I: An Introductionto Game Theory

“In football, everything is complicated by the presence of the opposite team.” (French original: “Au football,tout est complique par la presence de l’equipe adverse.”)1

O U T L I N E

2.1 Introduction 25

2.2 Understanding a Strategic Game 26

2.3 The Invisible Hand and the Fallacyof Aggregation Again 27

2.4 How Not to Play a Game 292.4.1 Pareto Dominance 29

Pareto Optimality and Suboptimality 29

2.5 How to Play a Game 292.5.1 Dominance 292.5.2 Best Answers 30

2.5.3 Nash Equilibria 31

2.6 How Many Games CanWe Play? 31

2.7 Summary 31

Chapter References 32

Further Reading 32

Further Reading—Online 32

Exercises 32

2.1 INTRODUCTION

We all know the following situation: PersonA is about to enter a building through a nar-row doorway when person B attempts to leavethrough the same exit. Trying to show goodmanners, person A will politely step aside in

order to let person B pass—however, so willperson B, in her case to let person A pass.It only takes a moment for the both of themto realize that the other person is trying to letthem pass, A realizes that B stepped aside;B realizes that A is waiting for her to pass.So they will again enter the doorway at thesame time only to find that this attempt tocoordinate has also failed.1Attributed to Jean-Paul Sartre.

25The Microeconomics of Complex Economies.

DOI: http://dx.doi.org/10.1016/B978-0-12-411585-9.00002-6 © 2015 Elsevier Inc. All rights reserved.

We are faced with a situation that seems tobe both simple, almost trivial, and difficult toresolve, thus complex. There are only two indi-viduals involved (A and B), they can choosebetween only two options (to wait or to enterthe doorway), their objective is the same: toresolve the situation and pass the doorwayeither A first or B first—it does not even mat-ter who passes first.

When faced with the situation it may seemlike a game: choose a strategy and hope forsome luck. Depending on what the other per-son does you win (i.e., you are able to pass) oryou lose. And this idea is not at all wrong.Many interactive situations, in fact, most of thesocial sphere may be conceptualized as one ina family of strategic games, each with its ownrules. There are games in which the playersshare a common objective like in the presentexample and games where they try to exploiteach other. There are games that offer an obvi-ous best choice option to all those involvedand games where the resolution is tricky andfull of uncertainty (like in the present case).There are games with only two agents andonly two options per agent and those with mil-lions and millions of participants and an infi-nite continuum of possible strategies. Thereare games which may be played repeatedlyand games that only offer one single try.

It is obvious that in an interactive world, itis always a simplification to reduce a social sit-uation to one game. The game can merely be amodel; it is a way to analyze the structure thatlies beneath the social interaction, its possibili-ties and opportunities, the development pathsof the interaction, less likely and more likelyoutcomes. A word of warning should not beforegone: Oversimplification results in inaccu-rate models; assuming a strategic game as amodel (or as part of a model) implies a num-ber of other assumptions (such as rationaldecision making) that are not always justified.The skilled and careful use of strategic gamesin models of social interactions, however,

reveals a whole new layer in the social systemthat is investigated—the layer of strategies andtheir consequences. This chapter will givea basic introduction to the modeling with stra-tegic games, an introduction to game theory.More formal concepts and advanced methodswill be discussed in Chapter 8.

2.2 UNDERSTANDINGA STRATEGIC GAME

To distinguish our model games from thecommon sense term for recreational games,we may specify our models as strategic games.A strategic game is characterized by a set ofparticipants, a set of behavior options for each ofthe participants, and a set of rules as well as theinformation available to each of the participants.The participants in a strategic game are com-monly called agents; the options among whichan agent chooses are referred to as her strategies.

There are many possible forms of games;some of the more prominent ones, includingrepeated games and sequential games, will beexplained in Chapter 8. One form, however,is as universally applicable and powerful asa model as it is simple and easy to use asa theoretical concept: simultaneous 2-person2-strategy normal-form games. Common to thisgroup of strategic games is that the gameinvolves two agents each of which choosesbetween only two strategies and the choices areto be made simultaneously without any possi-bility for collusion between the two agents.

Player B

Pass Wait

Player A Pass

0 0

1 1

Wait 1

1 0

0

FIGURE 2.1 Anti-coordination game; payoffs of therow player (A) highlighted in bold face.

I. BASICS OF THE INTERDEPENDENT ECONOMY AND ITS PROCESSES

26 2. TOOLS I: AN INTRODUCTION TO GAME THEORY

The game introduced in Figure 2.1 is amember of this group. We can write it asshown in Figure 2.1. The strategies of player Aare given in the rows, those of player B in thecolumns; for 23 2 strategies there are four pos-sible outcomes for which the payoffs of bothplayers are given (see Box 2.1 for details on theconcepts utility and payoff in game theory).Payoffs of the row player (A) are given in thelower left (bold), those of the column player(B) are given in the upper right of the respec-tive field. This matrix notation is commonlyused to write strategic games.

The view obtained from writing the game inmatrix notation (Figure 2.1) is the following:As said, both agents try to pass through the door,one first, then the other. There are two solutionsthat allow to accomplish this end: A waits and Bpasses or B waits and A passes. There are alsotwo less fortunate outcomes namely one whereboth agents attempt to pass and another onewith both agents waiting for the other one topass and neither of them actually attempting tomove through the doorway.

Game theory offers a number of approachesto solve strategic games and make predictionsregarding outcomes. Before these methods arediscussed, however, the following section willillustrate that strategic games may be used tomodel a broad range of concepts and phenom-ena in interactive economics.

2.3 THE INVISIBLE HAND ANDTHE FALLACY OF AGGREGATION

AGAIN

In Chapter 1, we introduced two generalconcepts of economic modeling, the invisiblehand and the fallacy of aggregation. Both tellillustrative stories of the direct interaction ofrational agents. The invisible hand describes asituation where every agent chooses thesocially optimal option out of her incentive tomaximize individually. Say, every agent hasan option to contribute (costs: 1) or not to con-tribute (zero cost) to a public good andreceives three times her own contribution (ben-efit: 2 or 0 respectively) and twice the contribu-tion of the other player(s) (2 or 0 respectively);see Figure 2.2.

The concept of optimality in interactivesituations, specifically the Pareto optimum, hasbeen introduced in Chapter 1. It is easy to seethat the game will arrive at the socially opti-mal (and single Pareto-optimal) point (4,4).Consequently, this model justifies ignoring themicro-level of direct interactions (since every-one chooses equivalently and optimally). Adifferent story is told by the fallacy ofaggregation.

Say, as before, the agents choose whether tocontribute or not; but this time, they onlyreceive twice the payoff of their opponents.

BOX 2.1

ORD INAL AND CARD INAL UT I L I TY

In game theory, utility is commonly given as

a cardinal measure, i.e., specific (quantified) pay-

offs are stated and payoff and utility variables

are treated as standing for exact quantities.

This, however, is not required for normal-form

games in pure strategies. It would be sufficient

to be decidable if one option A is (given the

choices of the other players) preferred to another

option B, if instead B is preferred to A or if the

agent is indifferent between the two as long as

this preference order is consistent. This is known

as ordinal utility measure. For details on theoret-

ical requirements for game-theoretic preference

functions see Chapter 8.

I. BASICS OF THE INTERDEPENDENT ECONOMY AND ITS PROCESSES

272.3 THE INVISIBLE HAND AND THE FALLACY OF AGGREGATION AGAIN

We can see that the second strategy seems tobe strictly better (strictly dominant) against thefirst one, the most likely outcome would there-fore be (0,0) which of course is Pareto inferiorcompared to (1,1); see Figure 2.3. The messageconveyed by this metaphor is that it is possiblethat agents choose a socially suboptimal situa-tion and do so rationally. Social interactionsmay take any form out of an extensive contin-uum of game structures with different proper-ties. Though it should be carefully chosenwhich game structure applies, game theory asa method is extremely powerful when consid-ering socioeconomic systems. To quote fromone of the most important recent game theorytextbooks, Hargreaves Heap and Varoufakis:“One is hard put to find a social phenomenonthat cannot be so described [as a strategicgame.]” (2004, p. 3).

The game given as an example for the meta-phor of the invisible hand is called a socialoptimum game; it is symmetric and for bothplayers the first strategy is preferable to the sec-ond no matter what the opponent does (a strictlydominant strategy). Both players choosing thisstrategy leads to a situation where neither of theplayers has an incentive to unilaterally deviatefrom this outcome which brings stability to

this result and justifies it being called anequilibrium—specifically a Nash equilibrium.This property (no incentive for unilateral devia-tion) does not hold for any of the alternativethree possible outcomes; the Nash equilibriumis thus unique and it is furthermore the onlyPareto-optimal outcome. The game discussedfor the fallacy of aggregation is a prisoners’dilemma; it is a symmetric game as well, it toocontains a strictly dominant strategy for bothplayers, leading to the single Nash equilibrium.However, this time the Nash equilibrium is theonly outcome that is not Pareto optimal.

Thus far, both games are 2-person, 2-strategynormal-form games; however, the same twogames may be defined as n-person games withthe same results. All n agents choose one of thetwo strategies. In order to make this choice,every agent will take into consideration the pos-sible choices of all the other agents. Say amongthe other agents, there are n1 who choose thefirst and n2 that opt for the second strategysuch that n1 1 n2 5 n2 1. In the social optimumgame, the payoffs resulting from the two strate-gies, Π1 and Π2, are as follows:

Π1 5211 31 2n1 5 21 2n1

Π2 5 01 01 2n1 5 2n1

That is, by choosing n1 the agent awardsherself and every other agent an additionalpayoff of 2. She will—as long as she isrational—find that this option is always prefer-able no matter how many other agents madethe same choice (i.e., no matter which value n1assumes). For the prisoners’ dilemma on theother hand, the payoffs resulting from the twostrategies are as follows:

Π1 5211 01 2n1 5211 2n1

Π2 5 01 01 2n1 5 2n1

It is easily seen that the second term, theone resulting from the contributions of otheragents, is equal in both games. However, thefirst, individual, term becomes negative in the

Player B

Strategy 1 Strategy 2

Player A Strategy 1

1 1

2 –1

Strategy 2 –1

2 0

0

FIGURE 2.3 Prisoners’ dilemma game.

Player B

Strategy 1 Strategy 2

Player A Strategy 1

4 4

2 2

Strategy 2 2

2 0

0

FIGURE 2.2 Social optimum game.

I. BASICS OF THE INTERDEPENDENT ECONOMY AND ITS PROCESSES

28 2. TOOLS I: AN INTRODUCTION TO GAME THEORY

prisoners’ dilemma. This does again notdepend on what the other agents do. No ratio-nal agent would therefore be willing to con-tribute by playing strategy 1 in this game.

Hence, provided that all agents are suffi-ciently informed and act rationally, we canpredict that a social optimum game will resultin all agents contributing as much as possible,while a game of the prisoners’ dilemma typewill prevent rational agents from contributingat all. As mentioned, the two games reflecttwo different if not oppositional economicprinciples, the invisible hand, which for morethan two centuries has guided classical andneoclassical thought, and the fallacy of aggre-gation, which describes a more complex, con-flictive, and interdependent setting.

Having discussed the game theory modelsof two radically different stories, of two oppo-site ways of conceptualizing economic reality,it has become clear that game theory is amethod. It is a powerful tool, perhaps the mostappropriate one to investigate socioeconomicsystems, but it is nothing more than that. Oneconcept may be modeled using game theoryjust as well as an entirely different one.

2.4 HOW NOT TO PLAY A GAME

2.4.1 Pareto Dominance

After detailing a number of illustrativeexamples in the previous sections, it is nowtime to proceed to the core concepts of gametheory. The above examples have shown thatthere are more preferable outcomes and lesspreferable outcomes. Some outcomes shoulddefinitely be avoided.

For instance, considering the anti-coordination game again, the agents willwant to avoid to be coordinated (i.e., beingstuck in the doorway or waiting for nothing).However, things are not always as easy: In theprisoners’ dilemma game (Figure 2.2), player B

wants to avoid the lower left field; player Adoes not agree—she is very much in favor ofgetting to exactly this field. Hence, it is imper-ative to obtain a neutral, general, and robustnotion which outcomes are bad for all of theinvolved and definitely to be avoided.

The concept is called Pareto dominance,named in honor of the Italian economistVilfredo Pareto. The concept recognizes thatpayoffs of different agents are fundamentallyuncomparable. Utility of agent A cannot betransformed into utility of agent B and viceversa. Still an outcome is still doubtlessly pref-erable if both agents improve or if one agentimproves while the other retains the same pay-off. Such an outcome is called Pareto superior;the other one is called Pareto dominated orPareto inferior.

Pareto Optimality and Suboptimality

A desirable outcome is one that is notPareto dominated by any other possible out-come. This may be illustrated as in Figure 2.4:in a figure with the payoffs of both agents onthe two axes, an outcome is Pareto dominatedif there is another outcome to the upper rightof this outcome. Outcome X is Pareto domi-nated by outcome Y. Outcomes Y, V, and W,however, are not Pareto dominated. Such anoutcome is called Pareto optimal or a Paretooptimum of the game.

A simple way of finding Pareto optima in astrategic game is to go through the possibleoutcomes one by one and cross every outcomeout that is Pareto dominated by another (anyother) one; see Figure 2.5.

2.5 HOW TO PLAY A GAME

2.5.1 Dominance

It has been discussed which outcomes aredesirable. The question now is: How do wemanage to achieve one of them?

I. BASICS OF THE INTERDEPENDENT ECONOMY AND ITS PROCESSES

292.5 HOW TO PLAY A GAME

Obviously in the prisoners’ dilemma gameabove, neither player would be inclined to playtheir second strategy. Any possible payoff theycould obtain with this strategy is less than thealternative given the other player’s choice. Sucha strategy is called a strictly dominated strategy.No rational agent will ever make this choice.A strategy for which every payoff (given thechoice of the opponent) is better than all alter-natives is called a strictly dominant strategy.

2.5.2 Best Answers

Returning to the anti-coordination game(Figure 2.1), we find that there are no domi-nant or dominated strategies in this game.

Hence, the concept of dominance is not helpfulin predicting outcomes.

Still, it is clear which outcomes are prefera-ble and thus to decide which strategy theagents choose given the choice of the otherplayer. This concept is termed the best answerstrategies. If a strategy 1 of player A leadsplayer B to choose her strategy 2 then strategy2 of B is called the best answer (or a bestanswer since there may be several strategieswith equal payoffs) to strategy 1 of A. Everyplayer has at least one best answer strategycorresponding to every choice of the otherplayer(s). It is helpful to identify all bestanswer strategies to all the strategies of all theplayers in a game: Underline the payoff theplayer hopes to achieve as a best response tothe choice(s) of the opponent(s) (Figure 2.6).

–2 –1 0 1 2 3

Payoff player A

Pay

off p

laye

r B

V

X

Y

W

–2

–1

0

1

2

3

X

X

X

X

FIGURE 2.4 Outcomes in prisoners’dilemma game (Figures 2.2 and 2.5) andPareto dominance: X is Pareto dominatedby Y; Y, V, and W are Pareto optimal.

Player B

Strategy 1 Strategy 2

Player A Strategy 1

1 1

2 –1

Strategy 2 –1

2 0

0

FIGURE 2.5 Pareto optima in a prisoners’ dilemmagame. (Crossed payoffs indicate Pareto dominated outcomes;all other outcomes are Pareto optima.)

Player B

Strategy 1 Strategy 2

Player A Strategy 1

0 0

11

Strategy 2 1

10

0

FIGURE 2.6 Best answer strategies (payoffs underlined)in an anti-coordination game.

I. BASICS OF THE INTERDEPENDENT ECONOMY AND ITS PROCESSES

30 2. TOOLS I: AN INTRODUCTION TO GAME THEORY

2.5.3 Nash Equilibria

An outcome that is achieved as a result ofevery player playing her best answer strategy iscalled a Nash equilibrium named in honor ofgame theorist and Nobel memorial laureate JohnNash. Every involved strategy is the best answerto the strategies of the other player(s), thus noplayer has an incentive to unilaterally deviatefrom the Nash equilibrium. It follows thata Nash equilibrium is—for rational players—an absorbing state, an outcome that is dynami-cally stable (if there were the option that one ormore players revise their choices), an outcomethat is more probable than others.

To obtain the set of Nash equilibria in astrategic game, underline the best answerpayoffs and identify the outcomes for whichall payoffs for all players are underlined.Figure 2.6 does this for the anti-coordinationgame discussed earlier in this chapter: theNash equilibria are the lower left and upperright fields, the outcomes in which the playerschoose different strategies.

2.6 HOW MANYGAMES CAN WE PLAY?

So far we restricted the considerations inthis chapter to a specific type of games: Gamesthat are limited to exactly one time periodin which all agents act simultaneously. Thesegames are referred to as normal-form games(because they are conveniently represented inthe matrix form, also called normal-form, asshown above). However, many different typesof games are conceivable. The agents maychoose strategies sequentially—in this case, thesecond player knows the decision of the firstone when she chooses her strategy (sequentialgames). Normal-form games may be playedrepeatedly, which means that the agentsremember choices and outcomes from previ-ous periods (repeated games). This allows

more complex strategies since they may con-tain instructions for more than one periodand since they may also take the opponents’actions and reactions into consideration.Repeated games may be repeated for a certaintime or indefinitely; they are also called super-games, particularly in case of indefinite repe-tition. These different types of games alsorequire refinements of some of the game-theoretic concepts; these will be detailed inChapter 8.

2.7 SUMMARY

This chapter offered a basic understandingof what a strategic game is, how it is used inmodeling interactive situations, and how thecentral concepts of basic game theory—bestanswers, dominance, Nash equilibria, andPareto optimality—are applied. Game theoryis only one in a number of methods that may beused to analyze and understand economicinteractions and their consequences. Until veryrecently, a large part of the profession ofeconomics chose to forego using game theory—though this is now slowly but steadily chang-ing. This was because direct interactions arevery difficult to fit into general equilibriummodels which in turn made it possible for thefirst time to analyze the economy as a wholeand the nontrivial interdependence betweendifferent sectors, still one of the most impor-tant accomplishments in the history of eco-nomics. However, this came at the cost ofsacrificing heterogeneity and direct interactionand reducing the social sphere to an agglomer-ation of homogeneous agents. The elaboratemodels of perfect markets in effect shifted theattention away from strategic interactions toanother part of economic reality. The marketperspective will be explained in Chapters 5through 7 while Chapter 8 and the followingchapters explain more advanced methods forthe analysis of interactive economies.

I. BASICS OF THE INTERDEPENDENT ECONOMY AND ITS PROCESSES

312.7 SUMMARY

A broad perspective on interactive econom-ics using and applying the methods explainedin this chapter will be detailed in Chapters 3and 4.

Chapter References

Hargreaves Heap, S.P., Varoufakis, Y., 2004. Game Theory:A Critical Introduction. Routledge, London, New York.

Further Reading

Binmore, K., 2007. Game Theory: A Very ShortIntroduction. Oxford University Press, Oxford.

Bowles, S., 2006. Social interactions and institutionaldesign. In: Bowles, S. (Ed.), Microeconomics: Behavior,Institutions, and Evolution. Princeton University Press,Princeton, NJ, pp. 23�55. , Chapter I.

Further Reading—Online

For further reading, see the textbook website at http://booksite.elsevier.com/9780124115859

EXERCISES

1. Identify:a. the Pareto optima in the anti-

coordination game (Figure 2.1) and inthe social optimum game (Figure 2.2)

b. the best answer strategies and Nashequilibria in the prisoners’ dilemma

(Figure 2.3) and the social optimumgame (Figure 2.2)

c. whether there are strictly dominantstrategies in the social optimum game(Figure 2.2)?

2. Consider the following game and identifyPareto optima, best answer strategies, andNash equilibria. Are there strictly dominantstrategies?

Player B

Strategy 1 Strategy 2

Player A Strategy 1

4 2

3 3

Strategy 2 1

1 2

4

3. How many Nash equilibria, how many bestanswer strategies, and how many Paretooptima does a 2-person 2-strategy normal-form game have at least? How many(of any of those concepts) does it have atmost? Consider as an example thefollowing game.

Player B

Strategy 1 Strategy 2

Player A Strategy 1

0 0

0 0

Strategy 2 0

0 0

0

I. BASICS OF THE INTERDEPENDENT ECONOMY AND ITS PROCESSES

32 2. TOOLS I: AN INTRODUCTION TO GAME THEORY